Capogna and Cowling showed that if φ is 1-quasiconformal on an open subset of a Carnot group G, then composition with φ preserves Q-harmonic functions, where Q denotes the homogeneous dimension of G. Then they combine this with a regularity theorem for Q-harmonic functions to show that φ is in fact C∞. As an application, they observe that a Liouville type theorem holds for some Carnot groups of step 2. In this article we argue, using the Engel group as an example, that a Liouville type theorem can be proved for every Carnot group. Indeed, the fact that 1-quasiconformal maps are smooth allows us to obtain a Liouville type theorem by applying the Tanaka prolongation theory.

Ottazzi, A., Warhurst, B. (2013). A Liouville Type Theorem for Carnot Groups: A Case Study. In M. Picardello (a cura di), Trends in Harmonic Analysis. Springer.

A Liouville Type Theorem for Carnot Groups: A Case Study

OTTAZZI, ALESSANDRO;
2013

Abstract

Capogna and Cowling showed that if φ is 1-quasiconformal on an open subset of a Carnot group G, then composition with φ preserves Q-harmonic functions, where Q denotes the homogeneous dimension of G. Then they combine this with a regularity theorem for Q-harmonic functions to show that φ is in fact C∞. As an application, they observe that a Liouville type theorem holds for some Carnot groups of step 2. In this article we argue, using the Engel group as an example, that a Liouville type theorem can be proved for every Carnot group. Indeed, the fact that 1-quasiconformal maps are smooth allows us to obtain a Liouville type theorem by applying the Tanaka prolongation theory.
Capitolo o saggio
Conformal map, Engel group, Tanaka prolongation.
English
Trends in Harmonic Analysis
Picardello, M
2013
Ottazzi, A., Warhurst, B. (2013). A Liouville Type Theorem for Carnot Groups: A Case Study. In M. Picardello (a cura di), Trends in Harmonic Analysis. Springer.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/38998
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